Ok lets stop right now to spread false informations.
A baron is first and foremost a mayor like any other mayor, it is the vassal of the count it is a mayor of and only and strictly that count and no one else.
Now, to be a baron, a mayor has to sign a contract with the duke of the count which it is a vassal of. That contract grant fundings to the mayor in exchange for military activities, including building and maintaining military structure, training, equiping and maintaining troops or any other duty the duke and the baron agree to in the contract, but, that contract does not change a single bit the relation between the baron and its sole and only lord, the count.
To go back to the OP's question.
First as a mayor, remember you have to be a mayor first in order to maybe become a baron, you will select a settlement, your "domain", i put domain in between quotes because as an aristocrat you do not have a domain, you only manage a settlement, so your "domain" extends only to the limits of your settlement.
But that settlement size only depends on the size of the settlement you select at domain selection and the work you put into it to develop it.
But again make no mistake, your mayoral power stops at the limit of your settlement, but then across the whole county starts your obligations toward the duke as stated in the contract you signed with it, which might include patrols, guard of key points, infrastructures or resources, work with the sheriff to maintain order, all or some or none of those.
Contrary to a wide spread misconception it is not in the duty of the baron to maintain law and order, that's the job of the sheriff. barons have the power to judge high crimes but they are not in charge of "police work", they are mayors with extra military duties.